Given $ m \angle RPS = 7x - 53$, $ m \angle QPR = 4x + 46$, and $ m \angle QPS = 103$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {4x + 46} + {7x - 53} = {103}$ Combine like terms: $ 11x - 7 = 103$ Add $7$ to both sides: $ 11x = 110$ Divide both sides by $11$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 7({10}) - 53$ Simplify: $ {m\angle RPS = 70 - 53}$ So ${m\angle RPS = 17}$.